### 3.1236 $$\int \frac{(b d+2 c d x)^2}{\sqrt{a+b x+c x^2}} \, dx$$

Optimal. Leaf size=75 $\frac{d^2 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c}}+d^2 (b+2 c x) \sqrt{a+b x+c x^2}$

[Out]

d^2*(b + 2*c*x)*Sqrt[a + b*x + c*x^2] + ((b^2 - 4*a*c)*d^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2
])])/(2*Sqrt[c])

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Rubi [A]  time = 0.0297058, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.115, Rules used = {692, 621, 206} $\frac{d^2 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c}}+d^2 (b+2 c x) \sqrt{a+b x+c x^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)^2/Sqrt[a + b*x + c*x^2],x]

[Out]

d^2*(b + 2*c*x)*Sqrt[a + b*x + c*x^2] + ((b^2 - 4*a*c)*d^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2
])])/(2*Sqrt[c])

Rule 692

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*d*(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1))/(b*(m + 2*p + 1)), x] + Dist[(d^2*(m - 1)*(b^2 - 4*a*c))/(b^2*(m + 2*p + 1)), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[
2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(b d+2 c d x)^2}{\sqrt{a+b x+c x^2}} \, dx &=d^2 (b+2 c x) \sqrt{a+b x+c x^2}+\frac{1}{2} \left (\left (b^2-4 a c\right ) d^2\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx\\ &=d^2 (b+2 c x) \sqrt{a+b x+c x^2}+\left (\left (b^2-4 a c\right ) d^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )\\ &=d^2 (b+2 c x) \sqrt{a+b x+c x^2}+\frac{\left (b^2-4 a c\right ) d^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c}}\\ \end{align*}

Mathematica [A]  time = 0.0551479, size = 71, normalized size = 0.95 $d^2 \left (\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{2 \sqrt{c}}+(b+2 c x) \sqrt{a+x (b+c x)}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*d + 2*c*d*x)^2/Sqrt[a + b*x + c*x^2],x]

[Out]

d^2*((b + 2*c*x)*Sqrt[a + x*(b + c*x)] + ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]
)/(2*Sqrt[c]))

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Maple [A]  time = 0.047, size = 108, normalized size = 1.4 \begin{align*} 2\,{d}^{2}cx\sqrt{c{x}^{2}+bx+a}+{d}^{2}b\sqrt{c{x}^{2}+bx+a}+{\frac{{b}^{2}{d}^{2}}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-2\,{d}^{2}\sqrt{c}a\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^2/(c*x^2+b*x+a)^(1/2),x)

[Out]

2*d^2*c*x*(c*x^2+b*x+a)^(1/2)+d^2*b*(c*x^2+b*x+a)^(1/2)+1/2*d^2*b^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)
)/c^(1/2)-2*d^2*c^(1/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^2/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.02853, size = 462, normalized size = 6.16 \begin{align*} \left [-\frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{c} d^{2} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) - 4 \,{\left (2 \, c^{2} d^{2} x + b c d^{2}\right )} \sqrt{c x^{2} + b x + a}}{4 \, c}, -\frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{-c} d^{2} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \,{\left (2 \, c^{2} d^{2} x + b c d^{2}\right )} \sqrt{c x^{2} + b x + a}}{2 \, c}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^2/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/4*((b^2 - 4*a*c)*sqrt(c)*d^2*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c)
- 4*a*c) - 4*(2*c^2*d^2*x + b*c*d^2)*sqrt(c*x^2 + b*x + a))/c, -1/2*((b^2 - 4*a*c)*sqrt(-c)*d^2*arctan(1/2*sqr
t(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - 2*(2*c^2*d^2*x + b*c*d^2)*sqrt(c*x^2 + b*x
+ a))/c]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} d^{2} \left (\int \frac{b^{2}}{\sqrt{a + b x + c x^{2}}}\, dx + \int \frac{4 c^{2} x^{2}}{\sqrt{a + b x + c x^{2}}}\, dx + \int \frac{4 b c x}{\sqrt{a + b x + c x^{2}}}\, dx\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**2/(c*x**2+b*x+a)**(1/2),x)

[Out]

d**2*(Integral(b**2/sqrt(a + b*x + c*x**2), x) + Integral(4*c**2*x**2/sqrt(a + b*x + c*x**2), x) + Integral(4*
b*c*x/sqrt(a + b*x + c*x**2), x))

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Giac [A]  time = 1.15866, size = 105, normalized size = 1.4 \begin{align*}{\left (2 \, c d^{2} x + b d^{2}\right )} \sqrt{c x^{2} + b x + a} - \frac{{\left (b^{2} d^{2} - 4 \, a c d^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2 \, \sqrt{c}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^2/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

(2*c*d^2*x + b*d^2)*sqrt(c*x^2 + b*x + a) - 1/2*(b^2*d^2 - 4*a*c*d^2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*sqrt(c) - b))/sqrt(c)